Question

Give a randomized incremental algorithm which given n circles in the plane, reports the point with greatest y coordinate that lies inside all of the circles.

Here, "the point with greatest y coordinate that lies inside all of the circles" means the point with greatest y coordinate among intersection points of all circles

greatet

Please give the pseudo-code! Thank you

Answer 1

Algo

1. take two circles and find their point of intersection if any.

to find point of intersection do the following

0

First calculate the distance d between the center of the circles. d = ||P₁ - P₀||.

• If d > r₀ + r₁ then there are no solutions, the circles are separate.
• If d < |r₀ - r₁| then there are no solutions because one circle is contained within the other.
• If d = 0 and r₀ = r₁ then the circles are coincident and there are an infinite number of solutions.

Considering the two triangles P₀P₂P₃ and P₁P₂P₃ we can write

a² + h² = r₀² and b² + h² = r₁²

Using d = a + b we can solve for a,

a = (r₀² - r₁² + d² ) / (2 d)

It can be readily shown that this reduces to r₀ when the two circles touch at one point, ie: d = r₀ + r₁

Solve for h by substituting a into the first equation, h² = r₀² - a²

So

P₂ = P₀ + a ( P₁ - P₀ ) / d

And finally, P₃ = (x₃,y₃) in terms of P₀ = (x₀,y₀), P₁ = (x₁,y₁) and P₂ = (x₂,y₂), is

x₃ = x₂ +- h ( y₁ - y₀ ) / d

y₃ = y₂ -+ h ( x₁ - x₀ ) / d

2. after finding P₃ put the point in equation of other n-2 circles to see if it lie inside, outside or on the circle

if valueevaluates to <0 inside circle

=0 on circle

>0 outside circle

3. for points inside or on (<=0) all n-2 circle store the y coordinate

4 repeat step 1,2 and 3 for all pairs of circle

5. find max of all y coordinate stored in step 3